Determine how many solutions exist for the system of equations. ${3x+y = 5}$ ${-18x+3y = -6}$
Solution: Convert both equations to slope-intercept form: ${3x+y = 5}$ $3x{-3x} + y = 5{-3x}$ $y = 5-3x$ ${y = -3x+5}$ ${-18x+3y = -6}$ $-18x{+18x} + 3y = -6{+18x}$ $3y = -6+18x$ $y = -2+6x$ ${y = 6x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -3x+5}$ ${y = 6x-2}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.